Editing Μ-law algorithm (section)

==Algorithm types==
The μ-law algorithm may be described in an analog form and in a quantized digital form.

===Continuous===
[[File:Mu-law function.svg|thumb|μ-law function|350x350px]]
[[File:Inverse mu-law function.svg|thumb|Inverse μ-law function|350x350px]]

For a given input {{mvar|x}}, the equation for μ-law encoding is<ref name="mulaw-equation">{{cite web |url=https://www.cisco.com/c/en/us/support/docs/voice/h323/8123-waveform-coding.html |title=Waveform Coding Techniques - Cisco |access-date=2020-12-07 |date=2006-02-02}}</ref>
<math display="block">
F(x) = \sgn(x) \dfrac{\ln(1 + \mu |x|)}{\ln(1 + \mu)}, \quad -1 \leq x \leq 1,
</math>

where {{math|1=''μ'' = 255}} in the North American and Japanese standards, and {{math|1=sgn(''x'')}} is the [[sign function]]. The [[Range of a function|range]] of this function is −1 to 1.

μ-law expansion is then given by the inverse equation:<ref name="mulaw-equation" />
<math display="block">
F^{-1}(y) = \sgn(y) \dfrac{(1 + \mu)^{|y|} - 1}{\mu}, \quad -1 \leq y \leq 1.
</math>

===Discrete===
The discrete form is defined in ITU-T Recommendation [[G.711]].<ref>{{cite web |url=http://www.itu.int/rec/dologin_pub.asp?lang=e&id=T-REC-G.711-198811-I!!PDF-E&type=items |title=ITU-T Recommendation G.711}}</ref>

G.711 is unclear about how to code the values at the limit of a range (e.g. whether +31 codes to 0xEF or 0xF0).{{citation needed|date=October 2016}}
However, G.191 provides example code in the [[C language]] for a μ-law encoder.<ref>{{Cite web|url=https://www.itu.int/rec/T-REC-G.191/en|title=G.191&nbsp;:&nbsp;Software tools for speech and audio coding standardization|website=www.itu.int}}</ref> The difference between the positive and negative ranges, e.g. the negative range corresponding to +30 to +1 is −31 to −2. This is accounted for by the use of [[1's complement]] (simple bit inversion) rather than [[2's complement]] to convert a negative value to a positive value during encoding.

{| class="wikitable"
|+ Quantized μ-law algorithm
! 14-bit binary linear input code !! 8-bit compressed code
|-
| +8158 to +4063 in 16 intervals of 256 || 0x80 + interval number
|-
| +4062 to +2015 in 16 intervals of 128 || 0x90 + interval number
|-
| +2014 to +991 in 16 intervals of 64 || 0xA0 + interval number
|-
| +990 to +479 in 16 intervals of 32 || 0xB0 + interval number
|-
| +478 to +223 in 16 intervals of 16 || 0xC0 + interval number
|-
| +222 to +95 in 16 intervals of 8 || 0xD0 + interval number
|-
| +94 to +31 in 16 intervals of 4 || 0xE0 + interval number
|-
| +30 to +1 in 15 intervals of 2 || 0xF0 + interval number
|-
| 0 || 0xFF
|-
| −1 || 0x7F
|-
| −31 to −2 in 15 intervals of 2 || 0x70 + interval number
|-
| −95 to −32 in 16 intervals of 4 || 0x60 + interval number
|-
| −223 to −96 in 16 intervals of 8 || 0x50 + interval number
|-
| −479 to −224 in 16 intervals of 16 || 0x40 + interval number
|-
| −991 to −480 in 16 intervals of 32 || 0x30 + interval number
|-
| −2015 to −992 in 16 intervals of 64 || 0x20 + interval number
|-
| −4063 to −2016 in 16 intervals of 128 || 0x10 + interval number
|-
| −8159 to −4064 in 16 intervals of 256 || 0x00 + interval number
|}